Exploring Marketing Research William G. Zikmund Chapter 17: Determining Sample Size

What does Statistics Mean? Descriptive Statistics Number of People Trends in Employment Data Inferential Statistics Make an inference about a population from a sample Copyright © 2000 by Harcourt, Inc. All rights reserved.

Population Parameter Versus Sample Statistics Copyright © 2000 by Harcourt, Inc. All rights reserved.

Copyright © 2000 by Harcourt, Inc. All rights reserved. Population Parameter Variables in a population Measured characteristics of a population Greek lower-case letters as notation Copyright © 2000 by Harcourt, Inc. All rights reserved.

Copyright © 2000 by Harcourt, Inc. All rights reserved. Sample Statistics Variables in a sample Measures computed from data English letters for notation Copyright © 2000 by Harcourt, Inc. All rights reserved.

Copyright © 2000 by Harcourt, Inc. All rights reserved. Making Data Usable Frequency Distributions Proportions Central Tendency Mean Median Mode Measures of Dispersion Copyright © 2000 by Harcourt, Inc. All rights reserved.

Copyright © 2000 by Harcourt, Inc. All rights reserved. Frequency Distribution of Deposits Frequency (number of people making deposits Amount in each range) less than $3,000 499 $3,000 - $4,999 530 $5,000 - $9,999 562 $10,000 - $14,999 718 $15,000 or more 811 3,120 Copyright © 2000 by Harcourt, Inc. All rights reserved.

Copyright © 2000 by Harcourt, Inc. All rights reserved. Percentage Distribution of Amounts of Deposits Amount Percent less than $3,000 16 $3,000 - $4,999 17 $5,000 - $9,999 18 $10,000 - $14,999 23 $15,000 or more 26 100 Copyright © 2000 by Harcourt, Inc. All rights reserved.

Copyright © 2000 by Harcourt, Inc. All rights reserved. Probability Distribution of Amounts of Deposits Amount Probability less than $3,000 .16 $3,000 - $4,999 .17 $5,000 - $9,999 .18 $10,000 - $14,999 .23 $15,000 or more .26 1.00 Copyright © 2000 by Harcourt, Inc. All rights reserved.

Measures of Central Tendency Mean - Arithmetic Average µ, population; , sample Median - Midpoint of the Distribution Mode - the Value that occurs most often Copyright © 2000 by Harcourt, Inc. All rights reserved.

Copyright © 2000 by Harcourt, Inc. All rights reserved. Population Mean Copyright © 2000 by Harcourt, Inc. All rights reserved.

Copyright © 2000 by Harcourt, Inc. All rights reserved. Sample Mean Copyright © 2000 by Harcourt, Inc. All rights reserved.

Copyright © 2000 by Harcourt, Inc. All rights reserved. Number of Sales Calls Per Day by Salespersons Number of Salesperson Sales calls Mike 4 Patty 3 Billie 2 Bob 5 John 3 Frank 3 Chuck 1 Samantha 5 26 Copyright © 2000 by Harcourt, Inc. All rights reserved.

Copyright © 2000 by Harcourt, Inc. All rights reserved. Sales for Products A and B, Both Average 200 Product A Product B 196 150 198 160 199 176 199 181 200 192 200 200 200 201 201 202 201 213 201 224 202 240 202 261 Copyright © 2000 by Harcourt, Inc. All rights reserved.

Measures of Dispersion The Range Standard Deviation Copyright © 2000 by Harcourt, Inc. All rights reserved.

Measures of Dispersion or Spread Range Mean absolute deviation Variance Standard deviation Copyright © 2000 by Harcourt, Inc. All rights reserved.

The Range as a Measure of Spread The range is the distance between the smallest and the largest value in the set. Range = largest value – smallest value Copyright © 2000 by Harcourt, Inc. All rights reserved.

Copyright © 2000 by Harcourt, Inc. All rights reserved. Deviation Scores The differences between each observation value and the mean: Copyright © 2000 by Harcourt, Inc. All rights reserved.

Copyright © 2000 by Harcourt, Inc. All rights reserved. Low Dispersion Verses High Dispersion 5 4 3 2 1 Low Dispersion Frequency 150 160 170 180 190 200 210 Value on Variable Copyright © 2000 by Harcourt, Inc. All rights reserved.

Copyright © 2000 by Harcourt, Inc. All rights reserved. 5 4 3 2 1 High dispersion Frequency 150 160 170 180 190 200 210 Value on Variable Copyright © 2000 by Harcourt, Inc. All rights reserved.

Copyright © 2000 by Harcourt, Inc. All rights reserved. Average Deviation Copyright © 2000 by Harcourt, Inc. All rights reserved.

Mean Squared Deviation Copyright © 2000 by Harcourt, Inc. All rights reserved.

Copyright © 2000 by Harcourt, Inc. All rights reserved. The Variance Copyright © 2000 by Harcourt, Inc. All rights reserved.

Copyright © 2000 by Harcourt, Inc. All rights reserved. Variance Copyright © 2000 by Harcourt, Inc. All rights reserved.

Copyright © 2000 by Harcourt, Inc. All rights reserved. The variance is given in squared units The standard deviation is the square root of variance: Copyright © 2000 by Harcourt, Inc. All rights reserved.

Sample Standard Deviation Copyright © 2000 by Harcourt, Inc. All rights reserved.

The Normal Distribution Normal Curve Bell Shaped Almost all of its values are within plus or minus 3 standard deviations I.Q. is an example Copyright © 2000 by Harcourt, Inc. All rights reserved.

Copyright © 2000 by Harcourt, Inc. All rights reserved. Normal Distribution Conventional Product Adoption Life Cycle: Five types of customers who will end up adopting a product INNOVATORS (2.5%): People who are the first to adopt a product. They are trend-setting, risk-taking, and are not typical consumers. Example: See a movie first weekend it’s out or in a preview. EARLY ADOPTERS (13.5%): People who are among the first but not as risk-taking. They adopt ideas early but with consideration, and they enjoy roles as opinion leaders. They spread the word about the product. Example: See a movie the first week of its release. EARLY MAJORITY (34%): Deliberate customers; adopt earlier than most customers but are not leaders. Example: See a movie after a few weeks, after reading all the reviews and getting recommendations from early adopters. LATE MAJORITY (34%): Skeptical customers, will only adopt an idea if the majority of people have tried it. Example: See a movie after it has been nominated for an Oscar. LAGGARDS (16%): Tradition-bound, suspicious of change; will adopt an idea only after it has been around long enough. Example: See a movie after it has come out on video. MEAN Copyright © 2000 by Harcourt, Inc. All rights reserved.

Copyright © 2000 by Harcourt, Inc. All rights reserved. Normal Distribution 13.59% 13.59% 34.13% 34.13% Conventional Product Adoption Life Cycle: Five types of customers who will end up adopting a product INNOVATORS (2.5%): People who are the first to adopt a product. They are trend-setting, risk-taking, and are not typical consumers. Example: See a movie first weekend it’s out or in a preview. EARLY ADOPTERS (13.5%): People who are among the first but not as risk-taking. They adopt ideas early but with consideration, and they enjoy roles as opinion leaders. They spread the word about the product. Example: See a movie the first week of its release. EARLY MAJORITY (34%): Deliberate customers; adopt earlier than most customers but are not leaders. Example: See a movie after a few weeks, after reading all the reviews and getting recommendations from early adopters. LATE MAJORITY (34%): Skeptical customers, will only adopt an idea if the majority of people have tried it. Example: See a movie after it has been nominated for an Oscar. LAGGARDS (16%): Tradition-bound, suspicious of change; will adopt an idea only after it has been around long enough. Example: See a movie after it has come out on video. 2.14% 2.14% Copyright © 2000 by Harcourt, Inc. All rights reserved.

Normal Curve: IQ Example 70 85 100 115 145 Copyright © 2000 by Harcourt, Inc. All rights reserved.

Standardized Normal Distribution Symetrical about its mean Mean identifies highest point Infinite number of cases - a continuous distribution Area under curve has a probability density = 1.0 Mean of zero, standard deviation of 1 Copyright © 2000 by Harcourt, Inc. All rights reserved.

Copyright © 2000 by Harcourt, Inc. All rights reserved. Standard Normal Curve The curve is bell-shaped or symmetrical About 68% of the observations will fall within 1 standard deviation of the mean About 95% of the observations will fall within approximately 2 (1.96) standard deviations of the mean Almost all of the observations will fall within 3 standard deviations of the mean Copyright © 2000 by Harcourt, Inc. All rights reserved.

Copyright © 2000 by Harcourt, Inc. All rights reserved. A Standardized Normal Curve Conventional Product Adoption Life Cycle: Five types of customers who will end up adopting a product INNOVATORS (2.5%): People who are the first to adopt a product. They are trend-setting, risk-taking, and are not typical consumers. Example: See a movie first weekend it’s out or in a preview. EARLY ADOPTERS (13.5%): People who are among the first but not as risk-taking. They adopt ideas early but with consideration, and they enjoy roles as opinion leaders. They spread the word about the product. Example: See a movie the first week of its release. EARLY MAJORITY (34%): Deliberate customers; adopt earlier than most customers but are not leaders. Example: See a movie after a few weeks, after reading all the reviews and getting recommendations from early adopters. LATE MAJORITY (34%): Skeptical customers, will only adopt an idea if the majority of people have tried it. Example: See a movie after it has been nominated for an Oscar. LAGGARDS (16%): Tradition-bound, suspicious of change; will adopt an idea only after it has been around long enough. Example: See a movie after it has come out on video. z 1 2 -2 -1 Copyright © 2000 by Harcourt, Inc. All rights reserved.

The Standardized Normal is the Distribution of Z Copyright © 2000 by Harcourt, Inc. All rights reserved.

Copyright © 2000 by Harcourt, Inc. All rights reserved. Standardized Scores Copyright © 2000 by Harcourt, Inc. All rights reserved.

Copyright © 2000 by Harcourt, Inc. All rights reserved. Standardized Values Used to compare an individual value to the population mean in units of the standard deviation Copyright © 2000 by Harcourt, Inc. All rights reserved.

Copyright © 2000 by Harcourt, Inc. All rights reserved. Linear Transformation of Any Normal Variable into a Standardized Normal Variable s s m X m Sometimes the scale is stretched Sometimes the scale is shrunk -2 -1 0 1 2 Copyright © 2000 by Harcourt, Inc. All rights reserved.

Copyright © 2000 by Harcourt, Inc. All rights reserved. Population Distribution Sample Distribution Sampling Distribution Copyright © 2000 by Harcourt, Inc. All rights reserved.

Copyright © 2000 by Harcourt, Inc. All rights reserved. Population Distribution Conventional Product Adoption Life Cycle: Five types of customers who will end up adopting a product INNOVATORS (2.5%): People who are the first to adopt a product. They are trend-setting, risk-taking, and are not typical consumers. Example: See a movie first weekend it’s out or in a preview. EARLY ADOPTERS (13.5%): People who are among the first but not as risk-taking. They adopt ideas early but with consideration, and they enjoy roles as opinion leaders. They spread the word about the product. Example: See a movie the first week of its release. EARLY MAJORITY (34%): Deliberate customers; adopt earlier than most customers but are not leaders. Example: See a movie after a few weeks, after reading all the reviews and getting recommendations from early adopters. LATE MAJORITY (34%): Skeptical customers, will only adopt an idea if the majority of people have tried it. Example: See a movie after it has been nominated for an Oscar. LAGGARDS (16%): Tradition-bound, suspicious of change; will adopt an idea only after it has been around long enough. Example: See a movie after it has come out on video. -s m s x Copyright © 2000 by Harcourt, Inc. All rights reserved.

Copyright © 2000 by Harcourt, Inc. All rights reserved. Sample Distribution Conventional Product Adoption Life Cycle: Five types of customers who will end up adopting a product INNOVATORS (2.5%): People who are the first to adopt a product. They are trend-setting, risk-taking, and are not typical consumers. Example: See a movie first weekend it’s out or in a preview. EARLY ADOPTERS (13.5%): People who are among the first but not as risk-taking. They adopt ideas early but with consideration, and they enjoy roles as opinion leaders. They spread the word about the product. Example: See a movie the first week of its release. EARLY MAJORITY (34%): Deliberate customers; adopt earlier than most customers but are not leaders. Example: See a movie after a few weeks, after reading all the reviews and getting recommendations from early adopters. LATE MAJORITY (34%): Skeptical customers, will only adopt an idea if the majority of people have tried it. Example: See a movie after it has been nominated for an Oscar. LAGGARDS (16%): Tradition-bound, suspicious of change; will adopt an idea only after it has been around long enough. Example: See a movie after it has come out on video. _ C X S Copyright © 2000 by Harcourt, Inc. All rights reserved.

Copyright © 2000 by Harcourt, Inc. All rights reserved. Sampling Distribution Conventional Product Adoption Life Cycle: Five types of customers who will end up adopting a product INNOVATORS (2.5%): People who are the first to adopt a product. They are trend-setting, risk-taking, and are not typical consumers. Example: See a movie first weekend it’s out or in a preview. EARLY ADOPTERS (13.5%): People who are among the first but not as risk-taking. They adopt ideas early but with consideration, and they enjoy roles as opinion leaders. They spread the word about the product. Example: See a movie the first week of its release. EARLY MAJORITY (34%): Deliberate customers; adopt earlier than most customers but are not leaders. Example: See a movie after a few weeks, after reading all the reviews and getting recommendations from early adopters. LATE MAJORITY (34%): Skeptical customers, will only adopt an idea if the majority of people have tried it. Example: See a movie after it has been nominated for an Oscar. LAGGARDS (16%): Tradition-bound, suspicious of change; will adopt an idea only after it has been around long enough. Example: See a movie after it has come out on video. Copyright © 2000 by Harcourt, Inc. All rights reserved.

Standard Error of the Mean Standard deviation of the sampling distribution Copyright © 2000 by Harcourt, Inc. All rights reserved.

Copyright © 2000 by Harcourt, Inc. All rights reserved. CENTRAL LIMIT THEORM Copyright © 2000 by Harcourt, Inc. All rights reserved.

Standard Error of the Mean Copyright © 2000 by Harcourt, Inc. All rights reserved.

Copyright © 2000 by Harcourt, Inc. All rights reserved.

Copyright © 2000 by Harcourt, Inc. All rights reserved. Parameter Estimates Point Estimates Confidence interval estimates Copyright © 2000 by Harcourt, Inc. All rights reserved.

Copyright © 2000 by Harcourt, Inc. All rights reserved. Confidence Interval Copyright © 2000 by Harcourt, Inc. All rights reserved.

Copyright © 2000 by Harcourt, Inc. All rights reserved.

Copyright © 2000 by Harcourt, Inc. All rights reserved.

Copyright © 2000 by Harcourt, Inc. All rights reserved.

Estimating the Standard Error of the Mean Copyright © 2000 by Harcourt, Inc. All rights reserved.

Copyright © 2000 by Harcourt, Inc. All rights reserved.

Random Sampling Error and Sample Size are Related Copyright © 2000 by Harcourt, Inc. All rights reserved.

Copyright © 2000 by Harcourt, Inc. All rights reserved. Sample Size Variance (Standard Deviation) Magnitude of Error Confidence Level Copyright © 2000 by Harcourt, Inc. All rights reserved.

Copyright © 2000 by Harcourt, Inc. All rights reserved. Sample Size Formula Copyright © 2000 by Harcourt, Inc. All rights reserved.

Copyright © 2000 by Harcourt, Inc. All rights reserved. Sample Size Formula Copyright © 2000 by Harcourt, Inc. All rights reserved.

Sample Size Formula - example Suppose a survey researcher, studying expenditures on lipstick, wishes to have a 95 percent confident level (Z) and a range of error (E) of less than $2.00. The estimate of the standard deviation is $29.00. Copyright © 2000 by Harcourt, Inc. All rights reserved.

Sample Size Formula - example Copyright © 2000 by Harcourt, Inc. All rights reserved.

Sample Size Formula - example Suppose, in the same example as the one before, the range of error (E) is acceptable at $4.00, sample size is reduced. Copyright © 2000 by Harcourt, Inc. All rights reserved.

Sample Size Formula - example Copyright © 2000 by Harcourt, Inc. All rights reserved.

Copyright © 2000 by Harcourt, Inc. All rights reserved. Calculating Sample Size 99% Confidence [ ] 1389 = 265 . 37 2 53 74 ú û ù ê ë é ) 29 )( 57 ( n 347 6325 18 4 Copyright © 2000 by Harcourt, Inc. All rights reserved.

Standard Error of the Proportion Copyright © 2000 by Harcourt, Inc. All rights reserved.

Confidence Interval for a Proportion Copyright © 2000 by Harcourt, Inc. All rights reserved.

Sample Size for a Proportion Copyright © 2000 by Harcourt, Inc. All rights reserved.